
We
parametrize
circle
Crl
a.
b)
with
Take
É=
☐
u
=
l¥u
.
2-
u
)
NO
)=
(
✗
101
,
YIO
)
)
i
Then
7.
É
=
V.
ou
=
"
+
÷
"
q
o
(
a.
b)
✗
(
O
)
=
At
r
↳
0
oggyzz
,
go
since
U
is
harmonic
.
YLO
)
=
btrstno
D=
f)
☐
•
(
Ux
.
Uyjdxdy
=
fux
dy
-
Uyetx
g-
'
(g)
=
f-
r
stall
,
r
↳
0
)
Blais
)
erla.hn
>
✗
Is
=
rdo
=
[
"
Uxfatrcoio
,
btrino
)
rooiodo
2k
f-
(
r
)
=
¥
,
§
UH
.Dds=÷fou(
atrwso.b-rsinordll-fokuyfa-rooo.b-rstnolrstnod02Brla.to
)
*
chain
rule
=
¥
/
Fula
+
rwso
.
btrstnos
do
=/
ok
(
Ulatrcoio
,
btrstnoi
)
do
Leibniz
rule
we
want
to
show
firs
=
0
.
=
¥r(
So
"
ulatrwso.btrs.no
)
do
)
*
Then
fer
]
=
constant
Recall
that
for
=T=
(
Fixx
.
⇐
ix.
y
,
)
,
=
(
zafer
)
)
f)
☐
•
Édxdy
=
§
F
,
dy
-
Fzdx
on
the
other
hand
,
Brlab
)
e
him
fer
,
=
u(
a.
b)
by
mean
value
theorem
and
continuity
of
UCX
.
4)
r→o
[
This
is
a
variation
of
"
Theorem
6
"
in
textbook
)
go
the
constant
is
Ula
.
b)
and
fer
)=
Ula
.
b)
☐